From the pic you posted, the terminals are TL, TM & TH. I'd read these as tank low, tank medium (or tank coMmon) and tank high 

, not WL or WH (well low or high). From the description on the controller front panel the lower LED is Source (well) Low and also labelled under current. From this I'd assume that the controller senses when the pump is dry as no water = no load = low current draw so the alarm comes up and (presumably) the motor is turned off, otherwise what's the point of the alarm ? Pretty easy to test. With the pump running lift it out of the water and see what happens. 
 
 
The Over Current alarm (with the high water level graphic) has me a bit confused but perhaps it assumes a ball valve in the tank which cuts off when the tank's full which then puts back pressure on the pump which makes it work harder = higher current. Don't really know, just guessing here. It would also come on if the screw was clogged with sand or whatever. Didn't you say that the old feller said the pump had its own built in protection ?
 
Just a few thoughts.
 
Deano

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